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Abstract

The subexponential decay observed in the γ-ray spectral maps of supernova remnants is explained in terms of tachyonic Cherenkov emission from a relativistic electron population. The tachyonic radiation densities of an electronic spinor current are derived, the total density as well as the transversal and longitudinal polarization components, taking account of electron recoil. Tachyonic flux quantization subject to dispersive and dissipative permeabilities is discussed, the matrix elements of the transversal and longitudinal Poynting vectors of the Maxwell–Proca field are obtained, Cherenkov emission angles and radiation conditions are derived. The spectral energy flux of an ultra-relativistic electron plasma is calculated, a tachyonic Cherenkov fit to the high-energy (1 GeV to 30 TeV) γ-ray spectrum of the Crab Nebula is performed, and estimates of the linear polarization degree are given. The spectral tail shows subexponential Weibull decay, which can be modeled with a frequency-dependent tachyon mass in the dispersion relations. Tachyonic flux densities interpolate between exponential and power-law spectral decay, which is further illustrated by Cherenkov fits to the γ-ray spectra of the supernova remnants IC 443 and W44. Subexponential spectral decay is manifested in double-logarithmic spectral maps as curved Weibull or straight power-law slope.

Keywords

Tachyonic γ-ray spectra of supernova remnants;

Crab Nebula, SNR IC 443, SNR W44;

Subexponential Weibull spectral decay;

Maxwell–Proca radiation fields;

Quantized tachyonic Cherenkov densities;

Transversal and longitudinal polarization

1. Introduction

We attempt a tachyonic Cherenkov interpretation of the high-energy GeV–TeV spectral peak of the Crab Nebula and of supernova remnants (SNRs) in general. There are currently an electromagnetic and a hadronic radiation mechanism in vogue to model the γ-ray spectra of SNRs, namely inverse Compton scattering and pion decay ( Bühler and Blandford, 2014, Abdo et al., 2010 and Ackermann et al., 2013). If the spectral tail of the remnant is curved, one uses an inverse-Compton fit resulting in exponential decay, whereas a power-law slope is viewed as evidence for pion decay and high-energy protons producing pions in collisions with heavier nuclei. Tachyonic Cherenkov spectra allow for a unified treatment, as they can interpolate between exponential and power-law spectral tails ( Tomaschitz, 2014), due to the frequency-dependent tachyon mass manifested by a subexponential decay factor in the energy flux.

We derive the quantized tachyonic Cherenkov densities generated by a freely propagating electron current in a permeable spacetime, the total radiation density as well as its transversal and longitudinal components. We average these densities over a relativistic electron plasma, calculate the spectral energy flux, and perform tachyonic Cherenkov fits to the high-energy spectrum of the Crab Nebula (Bühler and Blandford, 2014, Abdo et al., 2010, Aharonian et al., 2006, Abramowski et al., 2014 and Buehler et al., 2012) and the SNRs IC 443 and W44 (Ackermann et al., 2013, Albert et al., 2007, Acciari et al., 2009 and Giuliani et al., 2011). The frequency variation of the tachyon mass determines the decay of the energy flux, which is subexponential Weibull decay in the case of the Crab Nebula and a power-law slope for SNR IC 443 and SNR W44, even though the electron distributions are exponentially cut by their Boltzmann factor. The aim is to derive explicit formulas for the tachyonic radiation densities and put them to test by performing spectral fits to these remnants.

In Section 2, we outline the formalism of Maxwell–Proca radiation fields with negative mass-square in a dispersive and dissipative spacetime defined by complex frequency-dependent permeabilities. In Section 3, we discuss tachyonic flux quantization, starting with the discrete power coefficients of a free quantum current, and derive the matrix elements of the energy flux vectors by applying box quantization. In Section 4, we perform the continuum limit of the discrete power coefficients, obtaining in this way the quantized tachyonic Cherenkov densities. For each radiation frequency, there is a minimal Lorentz factor of the radiating charge which has to be exceeded for Cherenkov emission to occur at this frequency, and we explain how the emission angles of transversal and longitudinal quanta are related to this radiation condition.

In Section 5, we specialize the radiation densities to radiation from freely moving electrons, using the matrix elements of a Dirac current as radiation source. We derive the transversal and longitudinal radiation densities, which can be done quite explicitly without the need to specify the frequency dependence of the dispersive and absorptive permeabilities and the tachyon mass. The spectral densities are given in electronic energy/velocity parametrization as well as in Lorentz representation.

In Section 6, we average the tachyonic radiation densities over a relativistic electron distribution to model the spectral energy flux of supernova remnants. The GeV and TeV flux of the remnants depends on two decay factors. One is due to energy dissipation, the exponential being determined by the imaginary part of the wavenumbers defined by the complex dispersion relations. The second decay factor is a combination of the Boltzmann weight of the radiating electron population and the frequency-dependent tachyon mass and results in subexponential Weibull or power-law spectral decay of the flux densities. We calculate the transversal and longitudinal flux components and discuss their semiclassical and quantum limits and the effect of the longitudinal radiation on the transversal linear polarization degree. Fig. 1, Fig. 2 and Fig. 3 depict tachyonic Cherenkov fits to the γ-ray spectra of the Crab Nebula and the remnants IC 443 and W44, which admit subexponential spectral tails stretching over an extended energy range. In Section 7, we present our conclusions.

4.3. Transversal and longitudinal Cherenkov angles

The electronic wave vectors of the initial and final state are km=k0,mkmkm=k0,mkm and kn=k0,nknkn=k0,nkn, cf. after (4.3). We place these vectors into the (x,zx,z) plane and identify kmkm with the polar z axis. The electronic/tachyonic unit wave vectors can then be parametrized as k0,m=(0,0,1)k0,m=(0,0,1), k0,n=(sin⁡θn,0,cos⁡θn)k0,n=(sin⁡θn,0,cos⁡θn) and n=(sin⁡θ,0,cos⁡θ)n=(sin⁡θ,0,cos⁡θ). Energy–momentum conservation (4.7) gives

The classical tachyonic Cherenkov densities (Tomaschitz, 2014b) are recovered by performing the limit m→∞m→∞ in the quantum densities (5.5) and (5.6), so that the electron mass drops out; density pT(1)(ω,γ)pT(1)(ω,γ) vanishes in this limit.

The tachyonic energy flux produced by a thermal electron plasma can decay subexponentially (with Weibull exponent 0<ρ<10<ρ<1 in (6.9)), whereas inverse Compton scattering or electromagnetic synchrotron and curvature radiation result in exponential cutoffs if the cross-section or radiation density is averaged over a thermal or non-thermal electron population because of the exponential Boltzmann weight factor. Also the hadronic radiation theory of pion decay requires a non-exponential proton distribution in conjunction with a geometric fitting parameter to model the observed power-law decay and the extended cross-over region between the low- and high-frequency power-law slopes of the SNRs IC 443 and W44 (Ackermann et al., 2013). In contrast to electromagnetic Cherenkov radiation, the tachyonic Cherenkov flux admits a longitudinal polarization component, cf. (6.9) and Fig. 1, Fig. 2 and Fig. 3, due to the negative mass-square of the radiation field (Tomaschitz, 2009, Tomaschitz, 2009a and Tomaschitz, 2010b). This longitudinal component also affects the transversal linear polarization degree, cf. (6.10) and (6.20), the radiation being linearly polarized even in the GeV range, in contrast to inverse Compton scattering.

Weaker-than-exponential spectral decay, either Weibull decay (indicated by slightly curved slopes in double-logarithmic spectral plots as in Fig. 1) or a straight power-law slope (as in Fig. 2 and Fig. 3) has also been detected in the MeV spectra of atmospheric γ-ray flashes ( Dwyer and Uman, 2014) and solar flares ( Ackermann et al., 2014 and Ajello et al., 2014), as well as in the GeV spectra of γ-ray pulsars ( Abdo et al., 2013). Subexponential Weibull and power-law spectral tails can only extend over a finite energy range, the ultimate decay is exponential. Section 6 gives an overview of the various limit cases (semiclassical and quantum, high- and low-frequency, high- and low-temperature) of the tachyonic flux densities (6.2) and (6.3). Spectral fits are performed in finite frequency intervals, and the asymptotic approximations to these densities enumerated in Section 6 hold uniformly in finite intervals. A tachyonic γ-ray spectrum consists of an ascending power-law slope, cf. (6.11) and (6.13), followed by a cross-over region whose actual extent and curvature are mainly determined by the rescaled electronic temperature parameter β. The cross-over is followed by subexponential Weibull decay (cf. (6.9) and (6.21)) or a power-law descent (6.12), terminating in exponential decay (6.19).